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The Pseudorandomness of Legendre Symbols Under the Quadratic-Residuosity Assumption

  • 2026
  • OriginalPaper
  • Chapter
Published in:
Authors
Henry Corrigan-Gibbs
David J. Wu
Published in
Theory of Cryptography
Publisher
Springer Nature Switzerland

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Abstract

This chapter delves into the pseudorandomness of Legendre symbols under the quadratic-residuosity assumption, focusing on a variant of Damgård's pseudorandom generator. It introduces the concept of Legendre signatures and proves their security under standard cryptographic assumptions. The text explores the technical overview of the quadratic-residuosity problem and presents a detailed analysis of the Legendre weak pseudorandom function. Additionally, it discusses the implications of these findings for cryptographic applications and highlights the differences between the secret-modulus and public-modulus settings. The chapter concludes with a discussion of related work and open questions in the field.
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previous chapter Unconditional Foundations for Supersingular Isogeny-Based Cryptography
next chapter Dimensional eion: Improving the  Attack with Decomposition in Higher Bases
Title
The Pseudorandomness of Legendre Symbols Under the Quadratic-Residuosity Assumption
Authors
Henry Corrigan-Gibbs
David J. Wu
Copyright Year
2026
Book
Theory of Cryptography

Print ISBN: 978-3-032-12295-7

Electronic ISBN: 978-3-032-12296-4

Copyright Year: 2026

DOI
https://doi.org/10.1007/978-3-032-12296-4_10
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